We note that in using the simple form (Equation (1)) to calculate the \(\chi ^2\)  values we need to assume that the
measurements of colors and magnitude uncertainties are roughly in Gaussian distribution and uncorrelated,
which may not be true in the real case. Although the measurement of individual flux does follow a Gaussian
distribution, the measurements of magnitude and color generally do not follow it. Note that the SDSS magnitude
is asinh magnitude [1]}, which introduces an extra complexity. Future efforts will be
needed to improve the \(\chi ^2\)  calculation by considering the exact distributions of colors and magnitude uncertainties,
though the result may not change significantly. On the other hand,
the colors of quasars appear to be
less correlated when comparing with stars, which can be easily observed from the color-color diagrams
in the optical and near-infrared bands [2]}, [3]}, [4]}. The uncertainties
of magnitudes have also been shown to be minimally correlated [5]}. Therefore, the assumptions of non-correlations
of colors and magnitude uncertainties are believed to be reasonable. [6]} also adopted a similar formulae
as in Equation (1) to calculate the \(\chi ^2\)  values (see their Equation (1)) by considering the constant scatters of the median
color-redshift relations.
